| L(s) = 1 | + 2-s + 2.56·3-s + 4-s + 2.56·6-s − 1.56·7-s + 8-s + 3.56·9-s − 11-s + 2.56·12-s + 3.56·13-s − 1.56·14-s + 16-s + 1.43·17-s + 3.56·18-s + 3·19-s − 4·21-s − 22-s + 23-s + 2.56·24-s + 3.56·26-s + 1.43·27-s − 1.56·28-s + 5.56·29-s + 3.12·31-s + 32-s − 2.56·33-s + 1.43·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.47·3-s + 0.5·4-s + 1.04·6-s − 0.590·7-s + 0.353·8-s + 1.18·9-s − 0.301·11-s + 0.739·12-s + 0.987·13-s − 0.417·14-s + 0.250·16-s + 0.348·17-s + 0.839·18-s + 0.688·19-s − 0.872·21-s − 0.213·22-s + 0.208·23-s + 0.522·24-s + 0.698·26-s + 0.276·27-s − 0.295·28-s + 1.03·29-s + 0.560·31-s + 0.176·32-s − 0.445·33-s + 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.026476781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.026476781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 0.876T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.810734863207654154890427233238, −8.797487376288889424845444885060, −8.266681787834792259339019496361, −7.33711651474233879174569660318, −6.52314487369491662348056424884, −5.49538749284749347854151815322, −4.34462704363512086366350163883, −3.30788722784853078312501883417, −2.94698308972557970500392855251, −1.57561527725012579569170153590,
1.57561527725012579569170153590, 2.94698308972557970500392855251, 3.30788722784853078312501883417, 4.34462704363512086366350163883, 5.49538749284749347854151815322, 6.52314487369491662348056424884, 7.33711651474233879174569660318, 8.266681787834792259339019496361, 8.797487376288889424845444885060, 9.810734863207654154890427233238
